*(English)*Zbl 1093.46022

The paper under review is an ambitious and significant survey of absolute-valued algebras and absolute-valuable Banach spaces. The paper contains a detailed presentation of the results on absolute-valued algebras. An absolute-valued algebra is a real or complex algebra $A$ endowed with a norm $\parallel \dot{\parallel}$ satisfying $\parallel xy\parallel =\parallel x\parallel \phantom{\rule{0.166667em}{0ex}}\parallel y\parallel $ for all $x,y$ in $A$. A classical theorem due to S. Mazur establishes that there are only three absolute-valued associative real algebras ($\mathbb{R},\u2102$ and $\mathbb{H}$). However, when associativity is removed, absolute-valued algebras arise in many examples. Here we have a couple of illustrating examples appearing in the paper: every complete normed algebra is isometrically algebra-isomorphic to a quotient of a complete absolute-valued algebra, and every Banach space is linearly isometric to a subspace of a complete absolute-valued algebra.

The paper begins by reviewing the early works of A. Ostrowski, S. Mazur, A. A. Albert and F. B. Wright, and finishes leading us to the most recent developments of this theory, including many contributions due to the author. The so-called “inflexion point” of the theory, that is, the paper of K. Urbanik and F. B. Wright, is reviewed in depth, with all its consequences. In sections §2 and §3, the author reviews the important contributions to this theory made by K. Urbanik and M. L. El–Mallah. The paper also contains some new results and several new and alternative proofs of previously known results. Section §5 deals with those Banach spaces which underlie complete absolute-valued algebras, reviewing some recent results obtained by the author of the present survey in collaboration with J. Becerra and A. Moreno.

The paper contains a complete list of references on this topic, including Ph.D. theses which are completely devoted to absolute-valued algebras. The list of references contains also some previous surveys on the subject which are not easily available and not written in English. The present survey allows us to gain access to these previous results.